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Chapter 0: Problem 1
For exercises 1-12, simplify. $$ \frac{14}{32} $$
Short Answer
Expert verified
The simplified form of \(\frac{14}{32}\) is \(\frac{7}{16}\).
Step by step solution
01
Identify the Numerator and Denominator
The fraction given is \(\frac{14}{32}\). Here, 14 is the numerator and 32 is the denominator.
02
Find the Greatest Common Divisor (GCD)
Determine the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 14 and 32 is 2.
03
Divide Numerator and Denominator by GCD
Divide both the numerator and the denominator by their GCD. So, \(\frac{14 \div 2}{32 \div 2} = \frac{7}{16}\).
04
Simplified Fraction
The simplified form of \(\frac{14}{32}\) is \(\frac{7}{16}\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) is a key concept in fraction simplification. It refers to the largest number that can evenly divide both the numerator and the denominator. To find the GCD, list the factors of both numbers and identify the largest common one.
For example:
* Factors of 14: 1, 2, 7, 14
* Factors of 32: 1, 2, 4, 8, 16, 32
The GCD of 14 and 32 is 2, which is the largest factor both numbers share.
Numerator
The numerator is the top number in a fraction. It indicates how many parts of a whole are being considered. In the fraction \(\frac{14}{32}\), the numerator is 14. When simplifying fractions, you divide the numerator by the GCD. For instance, \(14 \div 2 = 7\). So, in our simplified fraction, the new numerator is 7.
Denominator
The denominator is the bottom number in a fraction. It shows the total number of equal parts the whole is divided into. In \(\frac{14}{32}\), 32 is the denominator. Dividing the denominator by the GCD helps simplify the fraction. For our fraction, \(32 \div 2 = 16\). Thus, the new denominator in the simplified fraction is 16.
Simplifying Fractions
Simplifying fractions makes them easier to work with. Here's how you simplify a fraction:
- Identify the numerator and the denominator.
- Find the Greatest Common Divisor (GCD).
- Divide both the numerator and the denominator by the GCD.
For example: To simplify \(\frac{14}{32}\),
* First, note the numerator (14) and the denominator (32).
* The GCD of 14 and 32 is 2.
* Divide both by the GCD: \(\frac{14 \div 2}{32 \div 2} = \frac{7}{16}\).
So, \(\frac{14}{32}\) simplifies to \(\frac{7}{16}\). This process reduces the fraction to its simplest form.
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![Problem 1 For exercises 1-12, simplify. ... [FREE SOLUTION] (3)](https://i0.wp.com/website-cdn.studysmarter.de/sites/14/2023/12/Rectangle-22721-2.png "Problem 1 For exercises 1-12, simplify. ... [FREE SOLUTION] (3)")
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![Problem 1 For exercises 1-12, simplify.
... [FREE SOLUTION] (2024)](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII=)